Question

F(x)=x^(3)+10x^(2)+33x+36

factor completley with work please

Answer 1

`\sf f(x) = (x + 3)(x + 3)(x + 4)`

Explanation:

To factor the cubic polynomial
`\sf f(x) = x^3 + 10x^2 + 33x + 36`, we can use various methods.

In this case, I'll use synthetic division to find one factor and then factorize the quadratic remaining.

1. First, we can try to find a root by testing potential factors of the constant term (36) divided by potential factors of the leading coefficient (1).

Let's try
`\sf x = -1` because it often makes calculations easier.

Synthetic Division:

The remainder is 12, not 0.

So,
`\sf x = -1` is not a root.

2. Next, let's try
`\sf x = -2`:

Again, the remainder is not 0.

3. Let's try
`\sf x = -3`:

We find that
`\sf x = -3` is a root because the remainder is 0.

4. Now, we can factor the polynomial as
`\sf f(x) = (x + 3)(x^2 + 7x + 12)`.

5. Further factor the quadratic term
`\sf x^2 + 7x + 12` by factoring it into:

`\sf x^2 + (4+3)x + 12`

`\sf x^2 + 4x + 3x + 12`

`\sf x(x+4)+3(x+4)`

`\sf (x+4)(x-3)`

`\sf (x + 3)(x + 4)`.

So, the completely factored form of
`\sf f(x)` is
`\sf f(x) = (x + 3)(x + 3)(x + 4)`.